Let $$A$$ be a set of $$n\left( { > 0} \right)$$ elements. Let $${N_r}$$ be the number of binary relations on $$A$$ and Let $${N_r}$$ be the number of functions from $$A$$ to $$A$$.
(a) Give the expression for $${N_r}$$ in terms of $$n$$.
(b) Give the expression for $${N_f}$$ in terms of $$n$$.
(c) Which is larger for all possible $$n, $$ $${N_r}$$ or $${N_f}$$?

Obtain the eigen values of the matrix $$$A = \left[ {\matrix{ 1 & 2 & {34} & {49} \cr 0 & 2 & {43} & {94} \cr 0 & 0 & { - 2} & {104} \cr 0 & 0 & 0 & { - 1} \cr } } \right]$$$

The function $$f\left( {x,y} \right) = 2{x^2} + 2xy - {y^3}$$ has

"If X then Y unless Z" is represented by which of the following formulas in propositional logic? (" $$\neg $$ " is negation, " $$ \wedge $$ " is conjunction, and " $$ \to $$ " is implication)